Question

A portfolio manager controls $10 million in common stock. In anticipation of a stock market decline, the decision is made to hedge the portfolio using the S&P 500 futures contract. The portfolio's beta is 1.1 and the dividend yield on this portfolio is 3% annually. One S&P 500 futures contract with 90 days to settlement date is traded at 1164.50. The S&P 500 index itself is at 1150.  (Use 365 days a year)

1. Calculate the number of futures contracts that should be bought or sold. Indicate whether you should buy or sell.
2. Suppose that when the contracts are closed out, the S&P 500 index has increased to 1380. Calculate the gain or loss on the combined positions (stock portfolio and futures contracts).
3. Suppose that when the contracts are closed out, the S&P 500 index has decreased to 1035. Calculate the gain or loss on the combined positions (stock portfolio and futures contracts).

Answer 1

1 future contract is for delivery of $250 times the index

Value of portfolio = $10 m = 10,000,000

Future price = 1164.50

Beta = 1.1

No. of future contracts (to be sold) = (1.1*10000000)/(250*1164.50) = 37.784

The futures contract should be sold. This is because in case the market falls, the value of portfolio decreases but the short futures contract gives a positive pay-off to offset the loss

(b) S&P index closes at 1380

% change in S&P = (1380-1150)/1150 = 20%

% change in Futures = 20% (same as change in S&P)

Pay-off from futures position = -37.784*250*(20%*1164.50) = -2199973.40

Return on portfolio = 1.1*20% + (3%/4) = 22.75%

Gain on portfolio = 10000000*22.75% = 2275000

Net gain or loss = 2275000+(-2199973.4) = $75,026.60 (gain)

(c)

S&P index closes at 1035

% change in S&P = (1035-1150)/1150 = -10%

% change in Futures = -10% (same as change in S&P)

Pay-off from futures position = -37.784*250*(-10%*1164.50) = 1099986.7

Return on portfolio = 1.1*(-10%) + (3%/4) = -10.25%

Loss on portfolio = 10000000*(-10.25%) = -1025000

Net gain or loss = -1025000+(1099986.7) = $74986.7 (gain)